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Overview

This tutorial focuses on pre-calculated outcome data meta-analysis using the R meta package.

Load Packages 

library(meta)
library(readxl)

Ratio Measures Meta-Analysis

Usefull Formulas
Excel Formula for Calculating the Odds Ratio (OR)

=(event.e / (n.e - event.e)) / (event.c / (n.c - event.c))

Excel Formula for Calculating the Standard Error log(OR)

=SQRT((1/event.e) + (1/(n.e - event.e)) + (1/event.c) + (1/(n.c - event.c)))

Excel Formula for Calculating the Risk Ratio (RR)

=(event.e / n.e) / (event.c / n.c)

Excel Formula for Calculating the Standard Error log(RR)

=SQRT((1/event.e) - (1/n.e) + (1/event.c) - (1/n.c))

Importing the Dataset

The entire dataset was converted into a dataframe format for easier handling and compatibility with the R example analysis. The dataframe structure allows seamless data manipulation and analysis using R’s built-in functions and packages.

ma$rr
##    Author Year  TE   seTE lower upper n.e n.c Subgroup_1 Variable_1
## 1 Study 1 2015 1.5     NA  0.95  2.38 150 150     Drug A         70
## 2 Study 2 2014 0.8     NA  0.47  1.35 120 130     Drug A         80
## 3 Study 3 2018 2.0     NA  1.35  2.96 200 210     Drug B         50
## 4 Study 4 2013 1.2 0.2087    NA    NA 180 190     Drug B         20
Performing the Meta-Analysis

To perform the meta-analysis of ratio measures we will use the metagen() function from the meta package. We need to provide some instructions for the metagen() function. These instructions are known as arguments, which can be one, two, three, or more inputs that the function uses to perform its task. The main arguments of the metagen() function are:

m.rr<-metagen(TE=log(TE), seTE=seTE, lower=log(lower), upper=log(upper),
               n.e=n.e, n.c=n.c,
               studlab = Author,
               data = ma$rr,
               method.tau = "REML",
               sm = "OR")
Summarizing the Results

To visualize the results of a meta-analysis conducted with the metagen() function, we will use the summary() function. It will generate a summary of the analysis:

summary(m.rr)
##             OR           95%-CI %W(common) %W(random)
## Study 1 1.5000 [0.9500; 2.3800]       22.8       24.4
## Study 2 0.8000 [0.4700; 1.3500]       17.3       21.8
## Study 3 2.0000 [1.3500; 2.9600]       31.2       27.3
## Study 4 1.2000 [0.7971; 1.8065]       28.7       26.5
## 
## Number of studies: k = 4
## Number of observations: o = 1330 (o.e = 650, o.c = 680)
## 
##                          OR           95%-CI    z p-value
## Common effect model  1.3806 [1.1088; 1.7191] 2.88  0.0039
## Random effects model 1.3335 [0.9242; 1.9241] 1.54  0.1240
## 
## Quantifying heterogeneity:
##  tau^2 = 0.0883 [0.0000; 2.0189]; tau = 0.2972 [0.0000; 1.4209]
##  I^2 = 63.0% [0.0%; 87.5%]; H = 1.64 [1.00; 2.83]
## 
## Test of heterogeneity:
##     Q d.f. p-value
##  8.11    3  0.0438
## 
## Details on meta-analytical method:
## - Inverse variance method
## - Restricted maximum-likelihood estimator for tau^2
## - Q-Profile method for confidence interval of tau^2 and tau
Creating the Forest Plot

To create the forest plot, we will use the forest() function. Alongside this function, we will include specific arguments to customize the appearance and content of the forest plot, such as labels, confidence interval formatting, and display options. These arguments ensure the plot is clear, informative, and tailored to the dataset being analyzed.

forest(m.rr,
       smlab = "Arrhythmia Recurrence",
       layout = "Revman5",
       sortvar = TE,
       lab.e = "Experimental", label.left = "Favors Experimental",
       lab.c = "Control", label.right = "Favors Control",
       ff.lr = "bold",
       leftcols = c("studlab", "Year", "n.e", "n.c", "w.random", "effect", "ci"),
       leftlabs = c("Studies", "Year", "Experimental\nTotal", "Control\nTotal", NA, NA, NA),
       text.random = "Random effects model",
       random = TRUE, 
       common = FALSE,
       test.overall.random = TRUE,
       rightcols = FALSE,
       colgap = "3mm",
       fs.heading = 12,
       fs.study = 12, 
       fs.hetstat = 10, 
       digits = 2,
       digits.pval = 2,
       pooled.totals = TRUE,
       col.square="darkcyan", col.square.lines="black",
       prediction = TRUE, col.predict = "#CEF2EE",col.predict.lines = "black", ff.predict = 1)

Difference Measures Meta-Analysis

Usefull Formulas
Excel Formula for Calculating the Mean Difference (MD)

=mean.e - mean.c

Excel Formula for Calculating the Standard Error MD

=SQRT(((sd.e)ˆ2/n.e) + ((sd.c)ˆ2/n.c))

Importing the Dataset

The entire dataset was converted into a dataframe format for easier handling and compatibility with the R example analysis. The dataframe structure allows seamless data manipulation and analysis using R’s built-in functions and packages.

ma$los
##    Author Year    TE   seTE lower upper n.e n.c Subgroup_1 Variable_1
## 1 Study 1 2015 -0.27     NA -0.62  0.08  30  30     Drug A         70
## 2 Study 2 2014  0.16 0.1173    NA    NA  50  50     Drug A         80
## 3 Study 3 2018 -0.24     NA -1.40  0.92  25  25     Drug B         50
## 4 Study 4 2013 -0.45     NA -0.79 -0.11 280 280     Drug B         20
Performing the Meta-Analysis

To perform the meta-analysis of difference measures we will use the metagen() function from the meta package. We need to provide some instructions for the metagen() function. These instructions are known as arguments, which can be one, two, three, or more inputs that the function uses to perform its task. The main arguments of the metagen() function are:

m.los<-metagen(TE=TE, seTE=seTE, lower=lower, upper=upper,
               n.e=n.e, n.c=n.c,
               studlab = Author,
               data = ma$los,
               method.tau = "REML",
               sm = "MD")
Summarizing the Results

To visualize the results of a meta-analysis conducted with the metagen() function, we will use the summary() function. It will generate a summary of the analysis:

summary(m.los)
##              MD             95%-CI %W(common) %W(random)
## Study 1 -0.2700 [-0.6200;  0.0800]       22.4       28.8
## Study 2  0.1600 [-0.0699;  0.3899]       51.9       34.9
## Study 3 -0.2400 [-1.4000;  0.9200]        2.0        7.0
## Study 4 -0.4500 [-0.7900; -0.1100]       23.7       29.3
## 
## Number of studies: k = 4
## Number of observations: o = 770 (o.e = 385, o.c = 385)
## 
##                           MD            95%-CI     z p-value
## Common effect model  -0.0890 [-0.2546; 0.0765] -1.05  0.2919
## Random effects model -0.1704 [-0.5077; 0.1668] -0.99  0.3219
## 
## Quantifying heterogeneity:
##  tau^2 = 0.0710 [0.0012; 0.8987]; tau = 0.2665 [0.0350; 0.9480]
##  I^2 = 69.8% [13.1%; 89.5%]; H = 1.82 [1.07; 3.09]
## 
## Test of heterogeneity:
##     Q d.f. p-value
##  9.93    3  0.0192
## 
## Details on meta-analytical method:
## - Inverse variance method
## - Restricted maximum-likelihood estimator for tau^2
## - Q-Profile method for confidence interval of tau^2 and tau
Creating the Forest Plot

To create the forest plot, we will use the forest() function. Alongside this function, we will include specific arguments to customize the appearance and content of the forest plot, such as labels, confidence interval formatting, and display options. These arguments ensure the plot is clear, informative, and tailored to the dataset being analyzed.

forest(m.rr,
       smlab = "Length of Stay",
       layout = "Revman5",
       sortvar = TE,
       lab.e = "Experimental", label.left = "Favors Experimental",
       lab.c = "Control", label.right = "Favors Control",
       ff.lr = "bold",
       leftcols = c("studlab", "Year", "n.e", "n.c", "w.random", "effect", "ci"),
       leftlabs = c("Studies", "Year", "Experimental\nTotal", "Control\nTotal", NA, NA, NA),
       text.random = "Random effects model",
       random = TRUE, 
       common = FALSE,
       test.overall.random = TRUE,
       rightcols = FALSE,
       colgap = "3mm",
       fs.heading = 12,
       fs.study = 12, 
       fs.hetstat = 10, 
       digits = 2,
       digits.pval = 2,
       pooled.totals = TRUE,
       col.square="darkcyan", col.square.lines="black",
       prediction = TRUE, col.predict = "#CEF2EE",col.predict.lines = "black", ff.predict = 1)

Working with different confidence intervals:

In meta-analyses, studies may report confidence intervals (CIs) with different levels of confidence (e.g., 90%, 95%, or 99%). To standardize the data for analysis, you must first calculate the Standard Error for all studies using their reported CIs, regardless of the confidence level. Once the SE is calculated, the treatment effect and its SE can be used uniformly in the metagen() function.

Excel Formula for Calculating the SE from 90% Confidence Interval - Studies with > 60 patients in each group
Ratio Measures

=(log(Upper_CI) - log(Lower_CI))/3.29

Difference Measures

=(Upper_CI - Lower_CI)/3.29

Excel Formula for Calculating the SE from 95% Confidence Interval - Studies with > 60 patients in each group
Ratio Measures

=(log(Upper_CI) - log(Lower_CI))/3.92

Difference Measures

=(Upper_CI - Lower_CI)/3.92

Excel Formula for Calculating the SE from 99% Confidence Interval - Studies with > 60 patients in each group
Ratio Measures

=(log(Upper_CI) - log(Lower_CI))/5.15

Difference Measures

=(Upper_CI - Lower_CI)/5.15

Calculating the SE - Studies with < 60 patients in each group

When the sample size in each group is small (e.g., fewer than 60 participants per group), confidence intervals should be calculated using the t-distribution instead of the normal distribution. The constants 3.92 , 3.29 , and 5.15 (derived from the normal distribution) must be replaced with larger values specific to the t distribution, accounting for the sample size and degrees of freedom (df).

  1. Before calculating the t statistic and subsequent SE for confidence intervals, it is essential to clearly define: (a) Confidence level; (b) Sample size in control (n.c) and experimental (n.e) group

  2. In an Excel cell, type the following formula using the variables: =tinv(1-CI, n.e+n.c-2)

  3. Calculate the SE Using the t-Statistic:

Ratio Measures

=(log(Upper_CI) - log(Lower_CI))/2*t

Difference Measures

=(Upper_CI - Lower_CI)/2*t

References:

  1. Core Team (2023). R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria. https://www.R-project.org/
  2. RStudio Team (2020). RStudio: Integrated Development for R. RStudio, PBC, Boston, MA URL http://www.rstudio.com/.
  3. meta package: Balduzzi S, Rücker G, Schwarzer G. How to perform a meta-analysis with R: a practical tutorial. In: Evidence-Based Mental Health. 22nd ed.; 2019:153-160.
  4. readxl package: Wickham H, Bryan J (2023). readxl: Read Excel Files. https://readxl.tidyverse.org, https://github.com/tidyverse/readxl.

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